Question

Using geometry, calculate the volume of the solid under z=9−x2−y2‾‾‾‾‾‾‾‾‾‾‾√ and over the circular disk x2+y2≤9.Using geometry, calculate the volume of the solid under z=9−x2−y2‾‾‾‾‾‾‾‾‾‾‾√ and over the circular disk x2+y2≤9.

Answer 1

54π/3

Explanation:

The disk in the problem is a disk with radius 3 that lies on the xy-plane, and the equation of z is the equation of a sphere centered at (0,0,0). This means that the picture we have with the given equations would be half of a sphere.

In order to solve this using geometry rather than calculus, we simply take the equation of the volume of a sphere and divide it by 2:

V(sphere)=(4/3)πr^3

V(half sphere)=(2/3)πr^3

Given that the radius is 3, we can plug this into the equation from above:

V(half sphere)=(2/3)π(3)^3

Now we can evaluate:

V(half sphere)=54π/3

Answer 2

= 54π/3

Explanation:

see the attached file